42 | New Scientist | 16 April 2022
Timothy Revell
is New Scientist’s
deputy US editorthat help mathematicians to build infinite sets
from the bottom up. Woodin instead started to
advocate a top-down approach that he called
ultimate L. This is a variant of an idea first
suggested by Gödel, and is akin to building an
observation deck high in the infinite tower to
help you see what is going on down below.
From above, ultimate L reaches the opposite
conclusion to the others – it says that the
continuum hypothesis is true.
Top down or bottom up, ultimate L, Martin’s
maximum or (*)? Each of these approaches has
its advantages and disadvantages, and it is up
to their proponents to demonstrate why their
idea is the right one. That is where one camp
appears to have just edged in front. In 2021,
Asperó and Schindler proved that if you are in
the bottom-up camp, you don’t have to choose
between Martin’s maximum and (*). With a
few technical tweaks, if you choose Martin’s
maximum as your additional axiom, that
implies (*) is a legitimate approach too.
The proof took the best part of a decade to
complete. Asperó and Schindler had originally
hit on the idea in 2011, but when they first
published their workings, Woodin spotted
a mistake. Yet the revised proof has certainly
rocked the world of mathematics. Not only
does it convincingly fuse two approaches,but the very fact that they turn out to be so
intertwined is seen by some as evidence that
this is the right answer – and that Cantor was
ultimately wrong in his conjecture. As Juliette
Kennedy at the University of Helsinki, Finland,
told Quanta magazine when the revised proof
emerged: “It’s one of the most intellectually
exciting, absolutely dramatic things that has
ever happened in the history of mathematics.”
“It’s remarkable,” says Woodin. “Typically,
when two completely different lines of
investigation converge on the same thing,
that’s taken as evidence for truth.” Despite
that, he still isn’t convinced, believing that
ultimate L is a neater and better way forward.
But his approach is still mired in the details.
Since he first proposed the idea in 2010,
he has yet to come up with the proof that
ultimate L really works as he thinks it should.
“I’m optimistic the proof is near,” he says.
“The trouble is the community is getting
very tired of me saying that. I’ve been
optimistic for several years now.”
In some ways, an answer to the continuum
hypothesis and the best way to firm up the
foundations of mathematics seems closer
than ever before. But it could just be an illusion.
That is the thing with the infinite: you tend
never to reach the end. ❚The real numbers – every number
that can be expressed as a decimal –
aren’t a countable set. This means
that however you pair them up with
the countable numbers, you will
always have some left over.
The proof goes something like
this. First, imagine creating a table
with all the countable numbers, 1,
2, 3, 4 and so on up to countable
infinity, and pairing them in a
second column with any randomly
chosen real number with an infinite
number of decimal places.
Now create a new number. Find
your first digit by adding 1 to the
first digit of the first real number
in your list, and your second
by adding 1 to the second digit
of the second real number and
so on. Here is an example:1 3 .153778425...
2 0. 7 36785323...
3 7.2 7 0286930...
4 42.3 4 603146...
... ...
The new number is 4.885...This process ensures our new
real number with infinitely many
decimal places has at least one digit
different from every number in the
(countably infinite) list, so can’t be
in it. This reveals a contradiction:
not every real number was on the
original list paired off with all
countable numbers. Therefore,
these two sets don’t have the same
size: there are more real numbers
than countable numbers.Accountably
uncountable
“ Two proofs converging on the same
answer suggests that answer is right”
Has time finally
been called on
the continuum
hypothesis?DA
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